Mr. T. Muir on Skeiv Determinants. 



393 



nant, the sign-factor necessary for perfect equivalence being 



(- 1 )"- 



These are self-evident propositions ; but when the order is 



the (2n + l)th, the determinant possesses a property which is 



not so readily apparent. For shortness in writing let us take 



a special case, the determinant of the 7th order, 



or A, say. 



If we increase each element of the first row by the corre- 

 sponding element of the last row, each element of the second 

 row by the corresponding element of the second row from the 

 end, and each element of the third row by the corresponding 

 element of the third row from the end, there results : — 



a 2 — a 6 a 3 —a b 

 ho— b R h — ba 



a x 



«2 



a 3 



a 4 



a 5 



«6 



a 7 



h 



h 



h 



K 



h 



h 



b 7 



C\ 



c 2 



C3 



c± 



c 5 



Ce 



c 7 



4 



d 2 



d 3 



0) 



-d 3 



—d 2 



-d x 



-*t! 



—c 6 



— c$ 



-c 4 



—H 



—c 2 



— 6l 



-&7 



-h 



-h 



-h 



-h 



-h 



-h 



— a 7 



— a 6 



— a 5 



— a i 



— a 3 



— a 2 



— a x 



A = 



% — a 7 

 bi — b 7 

 c x —c 7 

 di 

 -c 7 

 -h 7 

 —a 7 



C 2 Cq 



d 2 



-h 

 ; — a a 



C 3 C 5 



d 3 



-h 



■a 5 



a 5 —a 3 

 h — h 3 



—d 3 



— a 3 — 



^6 — a 2 



h 6 — b 2 



c&—c 2 



~d 2 



—c 2 



-h 2 



a 2 



a 7 — «! 



h 7 -h 



c 7 — c x 



—di 



—ci 



-h 

 — a-. 



Treating this determinant in the same way, but dealing with 

 columns instead of rows, we have 



A = 



a l — a 7 



C1-C7 

 -h 



—a 7 



a 2 — a 6 

 b 2 —b 6 



c 2 — c 6 

 d 2 



-he 

 — a 6 



a 3 —a h 

 b 3 — b 5 



C3 — C5 

 d 3 



— a b 





 

 



(O 



-c 4 





 

 

 



— C3— c 5 

 ~h 3 —b 5 

 — a 3 ~a 5 





 

 

 



— ^2 —Ce 

 —h 2 —b 6 



— a 2 — a 6 





 

 

 



-c l -c 7 



— b 1 —b 1 



— a 1 — a 1 



a i — a i a 2 — a e a $ — a 5 

 b\ — b 7 b 2 — b 6 b 3 —b 5 

 Ci—c 7 e 2 — c 6 c 3 — c 5 



XQ)X 



— C 3 —C 5 —C 2 —C 6 ~Ci— C7.(I 



— h 3 —b 5 — b 2 — b 6 — b l — b 1 

 —a 3 —a- —a 2 — ae — «i — a 7 



